Version 1
: Received: 15 March 2017 / Approved: 16 March 2017 / Online: 16 March 2017 (16:59:29 CET)
Version 2
: Received: 22 March 2017 / Approved: 22 March 2017 / Online: 22 March 2017 (04:48:57 CET)
How to cite:
Lander, A. Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Zeta Function to Re(s) = 1/2 and the Zeros of its Derivative to Re(s) > 1/2. Preprints2017, 2017030121. https://doi.org/10.20944/preprints201703.0121.v1
Lander, A. Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Zeta Function to Re(s) = 1/2 and the Zeros of its Derivative to Re(s) > 1/2. Preprints 2017, 2017030121. https://doi.org/10.20944/preprints201703.0121.v1
Lander, A. Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Zeta Function to Re(s) = 1/2 and the Zeros of its Derivative to Re(s) > 1/2. Preprints2017, 2017030121. https://doi.org/10.20944/preprints201703.0121.v1
APA Style
Lander, A. (2017). Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Z<em>eta</em> Function to Re(<em>s</em>) = 1/2 and the Zeros of its Derivative to Re(<em>s</em>) > 1/2. Preprints. https://doi.org/10.20944/preprints201703.0121.v1
Chicago/Turabian Style
Lander, A. 2017 "Two Finite Mirror-Image Series Restrict the Non-Trivial Zeros of Riemann’s Z<em>eta</em> Function to Re(<em>s</em>) = 1/2 and the Zeros of its Derivative to Re(<em>s</em>) > 1/2" Preprints. https://doi.org/10.20944/preprints201703.0121.v1
Abstract
Euler’s product formula over the primes and Euler’s zeta function equate to enshrine the Fundamental Theorem of Arithmetic that every integer > 1 is the product of a unique set of primes. The product formula has no zero, and with a domain ≤1 Euler’s zeta diverges. Dirichlet’s eta function η(s), negates alternate terms of zeta, permitting convergence when s∈C and Re(s) < 1, and its non-trivial zeros {ρ}, have a deep relationship with the distribution of the primes. The Riemann Hypothesis is that all the non-trivial zeros have Re(ρ) = 1/2. This work examines the symmetries in a partial Euler’s zeta series with a complex domain equating it to the difference between two finite vector series whose matched terms have mirror-image arguments, but whose magnitudes differ when Re(s) ≠ 1/2. Analytical continuation generates a modified eta series ηl(s), in which every lth term is multiplied by (1-l). If the integer l is appropriately determined by the Im(s), similar paired finite vector series have a difference that closely follows ηl (s) and their terminal vectors intersect in a unique way permitting zeros only when Re(s) = 1/2. Furthermore, those vectors tracking the derivatives of the series, have a special relationship permitting zeros of the differential only when Re(s) > 1/2.
Keywords
Riemann Hypothesis; Dirichlet eta function; Zeta function; prime numbers; number theory; critical-line; critical-strip; symmetry breaking
Subject
Computer Science and Mathematics, Mathematics
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Commenter: Anthony Lander
Commenter's Conflict of Interests: I am the author
https://www.amazon.co.uk/Symmetry-Zeros-Riemanns-Zeta-Function/dp/1986074145/ref=sr_1_1?s=books&ie=UTF8&qid=1520980905&sr=1-1&keywords=anthony+lander
The commenter has declared there is no conflict of interests.
MAYFEB Journal of Mathematics - ISSN 2371-6193
Vol 2 (2018) - Pages 1-81